Modeling and Inverse Problems in the Presence of Uncertainty Authors/Affiliations H. T. Banks, North Carolina State University, Raleigh, USA Shuhua Hu, North Carolina State University, Raleigh, USA W. Clayton Thompson, Noth Carolina State University, Raleigh, USA This book collects recent research—including the authors’ own substantial projects—on uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation itself. With many examples throughout addressing problems in physics, biology, and other areas, the book is suitable for applied mathematicians as well as scientists in biology, medicine, engineering, and physics. Key Features Reviews basic probability and statistical concepts, making the book self- contained Presents many applications and theoretical results from engineering, biology, and physics Covers the general relationship of differential equations driven by white noise (stochastic differential equations) and the ones driven by colored noise (random differential equations) in terms of their resulting probability density functions Describes the Prohorov metric framework for nonparametric estimation of a probability measure Contains numerous examples and end-of-chapter references to research results, including the authors’ technical reports that can be downloaded from North Carolina State University’s Center for Research in Scientific Computation Selected Contents Introduction. Probability and Statistics Overview. Mathematical and Statistical Aspects of Inverse Problems. Model Comparison Criteria. Estimation of Probability Measures Using Aggregate Population Data. Optimal Design. Propagation of Uncertainty in a Continuous Time Dynamical System. A Stochastic System and Its Corresponding Deterministic System. Frequently Used Notations and Abbreviations. Index. SAVE 20% SAVE 20% when you order online and enter Promo Code EZL18 FREE standard shipping when you order online. Catalog no. K21506 April 2014, 405 pp. ISBN: 978-1-4822-0642-5 $89.95 / £57.99
15
Embed
Modeling and Inverse Problems in the Presence of Uncertainty...Introduction. Probability and Statistics Overview. Mathematical and Statistical Aspects of Inverse Problems. Model Comparison
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Modeling and Inverse Problems in the Presence of Uncertainty
Authors/Affiliations
H. T. Banks, North Carolina State University, Raleigh, USA
Shuhua Hu, North Carolina State University, Raleigh, USA
W. Clayton Thompson, Noth Carolina State University, Raleigh, USA
This book collects recent research—including the authors’ own substantial projects—on uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation itself. With many examples throughout addressing problems in physics, biology, and other areas, the book is suitable for applied mathematicians as well as scientists in biology, medicine, engineering, and physics.
Key Features Reviews basic probability and statistical concepts, making the book self-
contained Presents many applications and theoretical results from engineering, biology,
and physics Covers the general relationship of differential equations driven by white noise
(stochastic differential equations) and the ones driven by colored noise (random differential equations) in terms of their resulting probability density functions
Describes the Prohorov metric framework for nonparametric estimation of aprobability measure
Contains numerous examples and end-of-chapter references to research results, including the authors’ technical reports that can be downloaded from North Carolina State University’s Center for Research in Scientific Computation
Selected Contents Introduction. Probability and Statistics Overview. Mathematical and Statistical Aspects of Inverse Problems. Model Comparison Criteria. Estimation of Probability Measures Using Aggregate Population Data. Optimal Design. Propagation of Uncertainty in a Continuous Time Dynamical System. A Stochastic System and Its Corresponding Deterministic System. Frequently Used Notations and Abbreviations. Index.
SAVE
20%
SAVE 20% when you order online and enter Promo Code EZL18 FREE standard shipping when you order online.
8.5 Biological Application: HIV Infection within a Host . . . . . 3368.5.1 Deterministic HIV Model . . . . . . . . . . . . . . . . 3368.5.2 Stochastic HIV Models . . . . . . . . . . . . . . . . . . 341
8.5.2.1 The Stochastic HIV Model Based on the BurstProduction Mode . . . . . . . . . . . . . . . 341
8.5.2.2 The Stochastic HIV Model Based on the Con-tinuous Production Mode . . . . . . . . . . . 343
8.5.3 Numerical Results for the Stochastic HIV Model Basedon the Burst Production Mode . . . . . . . . . . . . . 3438.5.3.1 Implementation of the Tau-Leaping
Algorithms . . . . . . . . . . . . . . . . . . . 3438.5.3.2 Comparison of Computational Efficiency of the
SSA and the Tau-Leaping Algorithms . . . . 3468.5.3.3 Accuracy of the Results Obtained by Tau-Leaping
a Fixed Delay . . . . . . . . . . . . . . . . . . . . . . . 3608.9.3 Comparison of the Stochastic Model with a Fixed Delay
and Its Corresponding Deterministic System . . . . . 3628.10 Simulation Algorithms for Stochastic Systems with Random
Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3628.11 Application in the Pork Production Network with a Random
Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3658.11.1 The Corresponding Deterministic System . . . . . . . 3658.11.2 Comparison of the Stochastic Model with a Random
Delay and Its Corresponding Deterministic System . . 3678.11.3 The Corresponding Constructed Stochastic System . . 3688.11.4 Comparison of the Constructed Stochastic System and
Its Corresponding Deterministic System . . . . . . . . 3708.11.5 Comparison of the Stochastic System with a Random
Delay and the Constructed Stochastic System . . . . . 3708.11.5.1 The Effect of Sample Size on the Comparison
of These Two Stochastic Systems . . . . . . 3718.11.5.2 The Effect of the Variance of a Random Delay
on the Comparison of These Two StochasticSystems . . . . . . . . . . . . . . . . . . . . . 375
Writing a research monograph on a “hot topic” such as “uncertainty prop-agation” is a somewhat daunting undertaking. Nonetheless, we decided tocollect our own views, supported by our own research efforts over the past12–15 years on a number of aspects of this topic, and summarize these forthe possible enlightenment they might provide (for us, our students and oth-ers). The research results discussed below are thus necessarily filled witha preponderance of references to our own research reports and papers. Innumerous references below (given at the conclusion of each chapter), we re-fer to CRSC-TRXX-YY. This refers to early Technical Report versions ofmanuscripts which can be found on the Center for Research in Scientific Com-putation website at North Carolina State University where XX refers to theyear, e.g., XX = 03 is 2003, XX = 99 is 1999, while the YY refers to thenumber of the report in that year. These can be found at and downloadedfrom http://www.ncsu.edu/crsc/reports.html where they are listed by year.
Our presentation here has an intended audience from the community of in-vestigators in applied mathematics interested in deterministic and/or stochas-tic models and their interactions as well as scientists in biology, medicine,engineering and physics interested in basic modeling and inverse problems,uncertainty in modeling, propagation of uncertainty and statistical modeling.
We owe great thanks to our former and current students, postdocs and col-leagues for their patience in enduring lectures, questions, feedback and someproofreading. Special thanks are due (in no particular order) to Zack Kenz,Keri Rehm, Dustin Kapraun, Jared Catenacci, Katie Link, Kris Rinnovatore,Kevin Flores, John Nardini, Karissa Cross and Laura Poag for careful read-ing of notes and suggested corrections/revisions on subsets of the material forthis monograph. However, in a sincere attempt to give credit where it is due,each of the authors firmly insists that any errors in judgment, mathematicalcontent, grammar or typos in the material presented in this monograph areentirely the responsibility of his/her two co-authors!!
We (especially young members of our research group) have been Generouslysupported by research grants and fellowships from US federal funding agenciesincluding AFSOR, DARPA, NIH, NSF, DED, and DOE. For this support andencouragement we are all most grateful.
H.T. BanksShuhua Hu
W. Clayton Thompson
xiii
Chapter 1
Introduction
The terms uncertainty quantification and uncertainty propagation havebecome so widely used as to almost have little meaning unless they are furtherexplained. Here we focus primarily on two basic types of problems:
1. Modeling and inverse problems where one assumes that a precise math-ematical model without modeling error is available. This is a standardassumption underlying a large segment of what is taught in many mod-ern statistics courses with a frequentist philosophy. More precisely, amathematical model is given by a dynamical system
dx
dt(t) = g(t,x(t), q) (1.1)
x(t0) = x0 (1.2)
with observation process
f (t; θ) = Cx(t; θ), (1.3)
where θ = (q,x0). The mathematical model is an n-dimensional deter-ministic system and there is a corresponding “truth” parameter θ0 =(q0,x00) so that in the presence of no measurement error the data can bedescribed exactly by the deterministic system at θ0. Thus, uncertaintyis present entirely due to some statistical model of the form
Y j = f(tj ; θ0) + Ej , j = 1, . . . , N, (1.4)
where f (tj ; θ) = Cx(tj ; θ), j = 1, . . . , N , corresponds to the observedpart of the solution of the mathematical model (1.1)–(1.2) at the jthcovariate or observation time and Ej is some type of (possibly statedependent) measurement error. For example, we consider errors that
include those of the form Ej = f(tj ; θ0)γ ◦ Ej where the operation γ◦
denotes component-wise exponentiation by γ followed by component-wise multiplication and γ ≥ 0.
2. An alternate problem wherein the mathematical modeling itself is amajor source of uncertainty and this uncertainty usually propagates intime. That is, the mathematical model has major uncertainties in itsform and/or its parametrization and/or its initial/boundary data, andthis uncertainty is propagated dynamically via some framework as yetto be determined.
1
2 Modeling and Inverse Problems in the Presence of Uncertainty
Before we begin the inverse problem discussions, we give a brief but usefulreview of certain basic probability and statistical concepts. After the probabil-ity and statistics review we present a chapter summarizing both mathematicaland statistical aspects of inverse problem methodology which includes ordi-nary, weighted and generalized least-squares formulations. We discuss asymp-totic theories, bootstrapping and issues related to evaluation of the correctnessof the assumed form of statistical models. We follow this with a discussionof methods for evaluating and comparing the validity of appropriateness ofa collection of models for describing a given data set, including statisticallybased model selection and model comparison techniques.
In Chapter 5 we present a summary of recent results on the estimation ofprobability distributions when they are embedded in complex mathematicalmodels and only aggregate (not individual) data are available. This is followedby a brief chapter on optimal design (what to measure? when and where tomeasure?) of experiments to be carried out in support of inverse problems forgiven models.
The last two chapters focus on the uncertainty in model formulation itself(the second item listed above as the focus of this monograph). In Chapter 7we consider the general problem of evolution of probability density functionsin time. This is done in the context of associated processes resulting fromstochastic differential equations (SDE), which are driven by white noise, andthose resulting from random differential equations (RDE), which are driven bycolored noise. We also discuss their respective wide applications in a numberof different fields including physics and biology. We also consider the generalrelationship between SDE and RDE and establish that there are classes ofproblems for which there is an equivalence between the solutions of the twoformulations. This equivalence, which we term pointwise equivalence, is inthe sense that the respective probability density functions are the same ateach time t. We show, however, that the stochastic processes resulting fromthe SDE and its corresponding pointwise equivalent RDE are generally notthe same in that they may have different covariance functions.
In a final chapter we consider questions related to the appropriateness ofdiscrete versus continuum models in transitions from small numbers of in-dividuals (particles, populations, molecules, etc.) to large numbers. Theseinvestigations are carried out in the context of continuous time Markov chain(CTMC) models and the Kurtz limit theorems for approximations for largenumber stochastic populations by ordinary differential equations for corre-sponding mean populations. Algorithms for simulating CTMC models andCTMC models with delays (discrete and random) are explained and simula-tions are presented for problems arising in specific applications.
The monograph contains illustrative examples throughout, many of themdirectly related to research projects carried out by our group at North CarolinaState University over the past decade.